Identification of the symmetry algebra as type C_N (sp_{2N})

Prove that the Lie algebra generated by the symmetry operators built from Σ^+, Σ^−, and their commutators (including the Cartan elements constructed from Σ^z, Λ^z, etc.) is isomorphic to the type-C Lie algebra C_N = sp_{2N}, with Cartan matrix entries matching the explicit matrix given in Section 3, and that the rank equals the number of sites N.

Background

After introducing Σ^±, the paper constructs additional operators (Σ^z, Λ^±, Λ^z, etc.) and uses Serre relations to infer the structure of the symmetry algebra. The conjecture asserts that this algebra is of type C_N.

The introduction explicitly conveys the conjecture that the operators generate the Lie algebra C_N, aligning with the detailed Cartan matrix presented later in Section 3.